Cats Tuesday: Gross-Out Charlie

The contents of Killer Charlie's stomach when he yakked outside:

• 1 backbone
  
• 2 skulls
  
• assorted front and rear legs
  
• a couple of itty bitty feet (nearly invisible)

And thank you for dropping by! Happy COT!

Technorati tags: Cats+on+Tuesday 

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Our Last Photowalk

During this photowalk, I again explored the lower beaver dam on May Brook, the century and a half old reservoir, and the old waterworks. Below I quote from Darlene Young's book A History of Barton Vermont:

Barton residents wanted a public water system for a number of compelling reasons. First and foremost, waterworks meant increased fire protection, making it possible to play hydrants throughout the village. Secondly, waterworks would supply an adequate amount of pure water to homes. And thirdly, a system of waterworks was a significant mark of a progressive village. Barton aspired to follow the lead of Newport and St. Johnsbury, which had already installed waterworks.

The talk in favor of a waterworks for Barton began in earnest in 1885. The village had improved its fire protection with a new fire engine and was building a series of cisterns for water storage. But many in the village saw the cisterns as only a temporary solution. The Orleans County Monitor pointed out that a system of waterworks would fill the need for fire protection as well as domestic use.

The trustees called a village meeting to consider the question of waterworks. The voters established a committee to investigate the feasibility and cost of a water system. The committee presented several options for supplying water to the village from Crystal Lake, estimating the cost at around $8,500. But voters rejected the proposal.

For the next few years, discussion of the question continued without any resolution. In 1886, the Legislature authorized the establishment of a system of waterworks for Barton. In 1888, the Legislature amended the water charter to permit the raising of $20,000 in bonds for the construction of the water system. Despite this, Barton voters took no immediate action.

In 1889, the village began looking at other water supplies besides Crystal Lake. Joel Foster of Montpelier, an engineer and veteran waterworks builder, cam to Barton several times to study the possibilities. Foster considered using a number of water sources: Roaring Brook, Parker Pond, and May Pond.

At a special village meeting held in July 1889, the first question put to the voters was a simple resolution. The resolution expressing the need for waterworks passed on a voice vote. The second article, "To see if the village would vote to erect a system of waterworks," brought a lengthy discussion. Foster explained the different routes and methods of bringing the water to the village. He estimated the cost at approximately $17,000. When debate ended, the voters, by a margin of 70 to 42, decided to use May Pond as their water source. They also approved a bond for $16,000.

Work began almost at once. Workmen built a stone dam at the reservoir, dug ditches, and laid the cast iron pipe to homes and businesses during the fall of 1889. Just before winter set in the village completed the mains and water began to flow.

If you wish, you can see the photos in my web gallery here, or in my Flickr set here.

Technorati tags: photowalk Barton Vermont Barton+Waterworks beaver+dam Barton+Mountain

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Carnival of Math #18: JD2718

Carnival of Math #18: JD2718

Infinity, fractal patterns in the Kalahari, the problems with calculators, T3 conferences (an idea that I always suspected!), college math instruction, the Fundamental Theorem of Arithmetic in sixth grade (I've done that!), the crossing number inequality (I have to do that), gifted and talented education (I went from Connecticut, where GAT was important, to Vermont, where GAT is ignored), a LOLCat?, NCLB assessments (my post!), polyhedra paper folding (a blogging friend), Diophantine equations and the products of digits (I participated!), another Pythagorean Theorem proof (got to check out that one), why is the US outta TIMSSA (we all know, right?), and many many other wonderful topics.

Click here now!
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Rock Climbing in Orleans

The Outdoors Club from Lake Region High School was learning how to climb rocks on a road cut in Orleans, Vermont at the intersection of Routes 5 and 58 on September 25, 2007. I was given permission to photograph and I was very careful to not get any faces. The entire set of photos can be seen in my web gallery here or in my Flickr set here.

Technorati tags: rock+climbing Orleans Vermont

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Calvin Coolidge and Plymouth Notch, Vermont

On September 23, I went to the Calvin Coolidge Homestead in Plymouth Notch, Vermont. Coolidge was the 30th president. You can find visitor information here. There was a Cheese and Harvest Festival going on, so it was crowded but fun. I posted a slideshow of the photos here. If you wish to view the individual photos and read the detailed comments, go to my web album here or to my Flickr set here.

I must return as soon as I can to view the cemetary and the Coolidge gravesite.

Technorati tags: Plymouth+Notch Vermont Calvin+Coolidge

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The Portland Montreal Pipe Line

::Portland Montreal Pipe Line:: About us

BG, a kind gentleman from Maine, e-mailed me about the Portland Montreal Pipe Line, whose warning sign I blogged about yesterday.  He sent me this link to the PMPL site. It's full of fascinating information: the pipeline opened in 1941 and is approximately 236 miles (or approximately 380 km) long.  It sends oil from Portland, Maine ("the largest oil port on the US East Coast") north to Montreal. It was built because of the need, during World War II, "to safely transport vital oil when enemy warships interrupted tanker traffic bound for refineries in Canada." I haven't heard of any problems with this pipeline. But when a problem happens, it will be a dozy. This is crude oil, and it's dangerous stuff. 

According to a Petroleum Administration Defense District map, this seems to be the only pipe line in New England.

Now I'm wondering if a hike along the pipeline (to Portland going south or to Montreal going north) is possible. Sounds interesting. There's a lot of history in this pipeline. Below is a very crude map, provided by PMPL, of the route the pipeline takes:

Thank you, BG, for the information.

Technorati tags: Portland+Montreal+Pipe+Line+Corporation crude+oil
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Dog Mountain Flowers In Autumn

Technorati tags: Dog+Mountain St.+Johnsbury Vermont autumn wildflowers 

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We Have Color

View from Dog Mountain, St. Johnsbury

Our color looked like this a couple of days ago but I think it's gone. It's hard to tell because it is pouring rain and I am all fogged in. I took this photo in St. Johnsbury where it has been much warmer and is further south than me. I never got a good photograph of our color. So this will have to be my "fall foliage" shot for this year. And the shot's not too good because the sky was overcast all day. This photo looks great when you click on it.

Technorati tags: Dog+Mountain fall+foliage St.+Johnsbury 

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What's My Spider Doing Now?

She is sitting on this pink fuzzy thing. She can't eat wasps: they are all gone. She is bigger than ever. Winter is coming and we had our first frost long ago. Does anybody know what she's up to?

Technorati tags: barn+spider

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WTF?

A sign found in the woods off of Duck Pond Rd, Barton

If I lived near this sign? You better believe I'd find a way to drill an itty bitty hole in this pipeline, attach a little narrow tube, and heat myself up this winter instead of staying close to 60°F (15.5°C). (I've been wanting to type WTF for months now. I finally got my chance.)

Technorati tags: oil pipeline

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Wal-Mart and the Battle of Vermont - The Boston Globe

A mask from the Bread and Puppet Museum, Glover.

Wal-Mart and the Battle of Vermont - The Boston Globe

Half the play is set in 2037, when only armed guerrillas from Vermont, led by dictators Ben and Jerry, are opposing the Wal-Martification of the United States. Instead of teaching children to watch television at School-Mart, a rebellious Vermont teacher instructs his charges how to make puppets, and so on.

I got a kick out of the reference to the Bread and Puppet. A few details of the history of Walmart in Vermont are wrong but it's interesting. Some folks are trying to get a Walmart into Derby so that we can get our underwear locally instead of in New Hampshire or Quebec.

It would be a crime for them to succeed. Why? Read Nickel and Dimed by Barbara Ehrenreich.

Technorati tags: Walmart Derby+Vermont Bread+and+Butter
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Photo Hunters: Curves

Thank you for visiting. Click Photohunt for more participants.

Technorati tags: PhotoHunt horse curves curvy
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I'm Pink for October, Too

For Dorothy, whom I admire so much.

I will be changing my text color (perhaps a few times until I find a pink that people can read).

Click this graphic, the badge on the right sidebar, or here so that you can go pink, too.

Technorati tags: Pink+For+October breast+cancer+awareness
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Dogs on Thursday: Bruno the Corn Maze Dog

Bruno is owned by the Corn Maze farmer. He is a corn-cob loving chocolate lab. He rips cornstalks out of the maze, shakes the cobs free, and then runs like a crazed chocolate lab with his corn. He eats several ears of corn a day! I caught him in this photo looking guilty after ripping up some cornstalks.

Technorati tags: O'Donnell's+Corn+Maze Westfield Vermont Dogs+on+Thursday

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Wordless Wednesday: Corn Maze Quiz

Technorati tags: O'Donnell+Farm corn+maze cow+udders+quiz Wordless+Wednesday

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MathNotations: Products of Digits: Challenges for Everyone...: Part 2

Remember Dave Marain's Challenge? I posted this challenge a couple of days ago here. He quoted a riddle from a book, The Righteous Men by Sam Bourne:

Just men we are, our number few
Describable in digits two
We're halved if these do multiply
If we few perish then all must die.

Dave then posed a challenge:

(1) Determine a 2-digit positive integer the product of whose digits is one-half the integer. Now, it won't take you long to find such a number (once you get past the elliptical phrasing), but that's just to whet your appetite. The real challenge begins:

(2) Prove that your answer to (1) is unique, i.e., there is only one solution to the problem.

Here is my solution:

Since the solution is a two digit number, we have two "slots" to fill, one for the tens digit and one for the ones digit. The digit in the tens place will be multiplied by 10 and the digit in the ones place will be multiplied by 1.

I will call the first digit, in the tens place, a; and I call the 2nd digit, in the ones place, b.

Because both a and b must be positive integers (we cannot have parts of people or negative amounts of people), we can only consider positive values of b. We are also restricted to the numbers 1 through 9 because we can only use one digit in each place value location.

In order for the expression for the value of a to be positive, 2b must be greater than 10 (or, b must be greater than 5). This leaves us with only the numbers 6, 7, 8 and 9 to consider.

We can now eliminate the odd numbers (7 and 9) because the numerator of the expression for the value of a must be an even number because the denominator will always be even (because the difference of 5 and b is always doubled). If the denominator will always be even, then the numerator must be even in order to get an integer.

We are now left with only 6 and 8 as possible solutions for b:

If b = 6:

If b = 8:

Therefore, 6 is the only possible solution for b. If b = 6, a = 3. Therefore, the integer we are looking for is 36. Half of thirty-six is 18. The product of 3 and 6 is 18. There were 36 people.

My middle school students would undoubtedly solve this problem using a table in Excel. They have had it drilled into them to show three ways to solve a problem: table, graph and equation. In this case, the graph would not be a proof for the solution but it would be confirmation that their solution is correct (assuming that their calculations are correct). I am curious about what equation they would be able to come up with, if any.



In the chart above, I failed to skitch in the point where both graphs intersect. But our students know how to explore the Excel chart to find where they intersect. As expected, the graph reflects the table from which it was made.

I am considering giving them this problem as one of their portfolio problems. I hope I have proof-read the first solution so that it is clear and error-free.

Technorati tags: product+of+digits mathematics+challenge Dave+Marain

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Cats Tuesday: Pansy In Love

Pansy, my kitty with all of the skeletal deformities, spies Buddy sleeping.

All photos are by grandson, Wingnut. July 31, 2007

She cuddles up next to him and he washes her.

Pansy is in heaven. She trusts few and loves only one: Buddy.

When he comes in the house, she runs
(as much as she can run with her deformities) to him and chirps loudly.

She falls over on him and rolls on the floor next to him.

Buddy is very patient with her (they know who is needy, don't they?).

Pansy can't jump and play like the others. But she can cuddle.

Pansy is in love. I'm glad she loves Buddy. (No other cat loves him.)

Technorati tags: Cats+on+Tuesday

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Heads or Tails: Haystack Mountain, Westmore, Vermont

Haystack Mountain rises above Long Pond in Westmore, Vermont (8 miles from my house) on the August morning that I went to watch the loons.

Technorati tags: Haystack+Mountain Long+Pond Westmore Vermont

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Me Or My Maths: And Negative Powers

Me Or My Maths: And Negative Powers.: ". . . Hence, (a) [the multiplication cancellation laws] then tell us that no matter what a is, a^0 = 1 ."

My students are always asking me why this is true (they just won't accept the pattern of 10^2=100, 10^1=10 and 10^0=1). Beans also shows us why a^-k = 1/a^k. It won't take long to show students (in college, not 7th grade!) this. Maybe, on second thought, 7th graders would get it.

By the way, I found the above mentioned exponent post on Me Or My Maths because of  How Many Frenchmen Can’t Be Wrong?: Vlorbik on Math Ed.

≤≥ ≤≥ ≤≥ ≤≥ ≤≥ ≤≥

I have two ways to solve Dave Marain's challenge (see previous post) and have not had even one minute to do the third way. I may do it tonight because I suddenly have time because I never made it to a meeting tonight. Dave specifically prohibited two of my ways, but they are methods my middle school students would use. My third way, I hope, will be more "grown-up."

Technorati tags: exponent adding+zero negative+exponents

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MathNotations: Products of Digits: Challenges for Everyone...

MathNotations: Products of Digits: Challenges for Everyone...: "Just men we are, our number few Describable in digits two We're halved if these do multiply If we few perish then all must die. "

Dave continues:

(1) Determine a 2-digit positive integer the product of whose digits is one-half the integer. Now, it won't take you long to find such a number (once you get past the elliptical phrasing), but that's just to whet your appetite. The real challenge begins:

(2) Prove that your answer to (1) is unique, i.e., there is only one solution to the problem.

Comment: We're looking for more than an exhaustive search through all ninety 2-digit numbers or a programmed solution. The key to this and all of the remaining questions is to find an approach to solving a single equation which has 2 or more variables whose domain is the set of positive integers. Students are usually not introduced to solving such equations but they appear frequently on SATs and Math Contests. Because we are looking only for positive integer solutions, a standard algebraic approach must be supplemented with arithmetic concepts and testing of several possibilities. Number theorists refer to these as Diophantine equations.

I have the solution (or is it "a solution"?). Now I have to see it if it is the only solution. I will update this post as I proceed. Right now I have to get ready for school.

"This investigation was authored by Dave Marain." 

Technorati tags: products+of+digits math+challenge Dave+Marain
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